Completely non unitary contractions and Analyticity

We study completely non-unitary contractions $T$ with finite dimensional defect spaces $\mathcal{D}_T$ and $\mathcal{D}_{T^*}$. We present a complete classification of all such contractions $T$ that satisfy a generalized property of Hardy shift operator: $\mathcal{D}_T\subseteq\mathcal{D}_{T^*}$. We show that $T$ is analytic if and only if it has no non-zero eigenvalue. Furthermore, we characterize the hyponormality of all those $T$.